THE LOCAL RADIO–IR RELATION IN M51

M51 was observed with the VLA at 20 cm (1.4 GHz) in the A, B, C, and D configurations, at 6 cm (4.9 GHz) in the B, C, and D configurations and at 3.6 cm (8.4 GHz) in the C and D configurations. At 3.6 cm, the primary beam FWHM is 5 4, thus two pointings were necessary to cover the full disk of M51 with a size of 10'. The VLA data were taken between 1988 and 2005 as part of several projects, corresponding to a total integration time of ∼170 hr on-source spread over all three frequencies. Table 2 provides a summary of the different observation runs. For the two most compact configurations (C and D) the continuum correlator mode was used, while observations in A and B configurations were done in multichannel continuum mode, using the four intermediate frequency (IF) spectral line mode. Both setups use two IFs. The central frequencies and corresponding bandwidths are listed in Table 2. During all observations, the quasar 3C286 was observed for flux calibration. For the 20 cm observations in C configuration, the quasars 3C48, 3C138, and 3C138 were also available as additional flux calibrators. The different projects used the following sources as phase calibrators: 1335+457, 1349+536, 1409+524, 1219+484, and 137+434. Depending on the frequency and configuration, the phase calibrator was observed every 10–45 minutes.

Data reduction was done using standard AIPS routines. Each day of observation has been calibrated separately. We performed standard flux calibration and atmospheric correction for the uv-data of each observation day. Before and after those calibrations, the data were inspected for radio frequency interference (RFI) (for each IF and polarization) and flagged in consequence. For the data observed in the multichannel correlator mode (20 cm in A and B configurations and 6 cm in B configuration), calibration and flagging were done on the pseudo-continuum channel and then applied to the "Line" data. Bandpass calibration was also performed on the "Line" data using the flux calibrator and the pseudo-continuum channel. Finally, we extract the source from the calibrated data with the task SPLIT which produces single-source files and applies the calibration. All days of observation in one configuration were then combined together with the task DBCON. At this point, we have one fully calibrated single-source data set for each configuration and each frequency.

For each frequency, we combined the fully calibrated uv data in the different configurations. First, we converted the coordinates of the oldest uv data from the B1950 to J2000 coordinates system with the tasks REGRD and UVFIX (20 cm data in C and D configurations and 6 cm data in D configuration). Then, the IFs being different for the different projects, the IFs were splitted with the task SPLIT for each configuration, and then recombined with the task DBCON into a single-channel data set. In the case of uv data obtained in the multichannel continuum correlator mode (configurations A and B at 20 cm and 6 cm), this step was done after all channels were splitted and recombined into a single data set. Therefore, at each frequency, a single data set was obtained for all configurations which were then combined in chronological order (the older observations first) with DBCON to obtain a final uv data set at a given reference frequency.

In a next step, the radio maps are produced using the task IMAGR. As the 20 cm and 6 cm data cover a large field of view (FOV), we first divided the field into 13 and 55 facets, of 4096 × 4096 pixels and 2048 × 2048 pixels at 6 cm and 20 cm, respectively. Each facet at 20 cm and 6 cm (respectively, each pointing at 3.6 cm) was then imaged, with the multi-resolution CLEAN in AIPS (Rich et al. 2008; Greisen et al. 2009), using several circular Gaussian model sources, in order to optimally deconvolve and image the extended faint emission. At 6 cm and 3.6 cm, we choose a number of circular Gaussian models equal to the number of observed configurations: two Gaussian models at 3.6 cm, of width equal to 0'' (point source) and 63 (with a flux density cutoff of 35 and 40 mJy beam⁻¹, respectively), three Gaussian models at 6 cm, of width 0'', 63, and 20'' with flux density cutoffs of 0.05, 0.06, and 0.2 mJy beam⁻¹, respectively. At 20 cm, four Gaussian models of width 0'', 47, 15'', and 47'' were chosen (with a respective flux density cutoff of 0.03, 0.3, 3, and 30 mJy beam⁻¹). The widths of the Gaussians are multiple of the robust 0 beam size (24, 20, and 15 at 3.6 cm, 6 cm, and 20 cm, respectively), separated by a factor of $\sqrt{10}$, thus giving a factor of 10 between the beam area of successive Gaussians. For the data at 20 cm and 6 cm we used respectively a robust weighting of ROBUST =1 and ROBUST =0, and at 3.6 cm we deconvolved with uniform weighting for maximal resolution (ROBUST =−4). After the deconvolution, we combined the facets/pointings using the task FLATN to construct the final images at 20 cm, 6 cm, and 3.6 cm. This task performs also a correction for the primary beam attenuation while combining the two pointings at 3.6 cm. During this correction, pixels outside the 10% power radius of the primary beam were blanked.

At 6 cm and 3.6 cm, the largest scale structures detectable (5' and 3', respectively) are smaller than the size of the galaxy system (10'). This implies that at these wavelengths the VLA observations can miss extended emission from M51. To correct for missing short spacings, we use the same method as Tabatabaei et al. (2007a), which consists of combining the final VLA CLEANed images with single dish data obtained with the Effelsberg 100 m telescope (Fletcher et al. 2010). First, the VLA maps were convolved to a circular beam, 2'' at 6 cm and 24 at 3.6 cm, the corresponding Effelsberg beam size being 180'' and 90'', respectively. We subtracted strong point sources in the VLA images to avoid bias in the combined total flux and the VLA and Effelsberg maps were combined in the uv-plane with the task IMERG in AIPS. The only free parameter of this task is the choice of the uv range of the interferometric data that should be filled by the single-dish values. We choose the uv range such that the integrated flux densities of the resulting merged map and of the Effelsberg map are equal. The overlap uv range is 0.51–0.71 kλ at 6 cm and 1.01–1.33kλ at 3.6 cm. The VLA point sources were then added back to the short-spacing corrected (SSC) maps. The total flux densities of the resulted SSC and the corresponding single dish maps are in agreement within 0.1%. Short-spacing correction is not necessary at 20 cm, since observation from the compact D configuration of the VLA allows one to detect structures with sizes up of 15', larger than M51.

Finally, we correct the 20 cm and SSC 6 cm maps for primary beam attenuation with the task PBCOR, with the same cutoff value of 10% of the primary beam radius used for the 3.6 cm image. The final 3.6 cm SSC, 6 cm SSC, and 20 cm maps corrected from primary beam attenuation have 4096 × 4096 pixels with a pixel size of 04, 02, and 03 at 3.6 cm, 6 cm, and 20 cm, respectively, and a final resolution of 24 × 24 at 3.6 cm, 2'' × 2'' at 6 cm, and 14 × 13 at 20 cm. The rms noise level is similar for the 20 cm (σ_noise = 11 μ Jy beam⁻¹) and 6 cm data (σ_noise = 16 μ Jy beam⁻¹), while it is higher in the case of the 3.6 cm map (σ_noise = 25 μ Jy beam⁻¹).

For comparison with our high-resolution radio maps, we used the Spitzer MIPS 24 μm map from the fifth Spitzer Infrared Nearby Galaxies Survey (SINGS) data delivery⁵ and the 8 μm polycyclic aromatic hydrocarbon (PAH) emission map, which corresponds to a stellar continuum subtracted 8 μm image, created by Regan et al. (2006). The description of the observations and reduction can be found in Kennicutt et al. (2003) and Calzetti et al. (2005). Briefly, the data reduction of these images used the standard MIPS pipeline (Gordon et al. 2005). In addition, some extra steps were performed to improve the image quality. For the MIPS 24 μm image, flat fielding was first performed, hot and dead pixels were removed, and latent images from bright sources and the background were subtracted. CRs were removed while the mosaic was created. For the IRAC images additional steps include correction from geometric distortion and rotation, image offsets and bias drift, CR removal, and background subtraction. Finally the stellar continuum was subtracted (Regan et al. 2006). The final resolution at 24 μm is 57 (Figure 1, top left) and it is 2'' for the 8 μm PAH emission map (Figure 1, bottom left).

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As the MIPS point-spread function (PSF) has strong secondary lobes and Airy rings which would affect a direct comparison to the radio images, we applied the HiRes deconvolution algorithm (Backus et al. 2005) to the IR maps. In addition, this deconvolution allows for an increase of the resolution of up to a factor of two in FWHM (Velusamy et al. 2008). Therefore, we can analyze our radio and IR data and investigate the correlation between the two without degrading our radio data by convolution with the MIPS PSF as it has been done in a previous study (Murphy et al. 2006). HiRes is based on the maximum correlation method, an extension of the Lucy–Richardson deconvolution algorithm and has been well tested on IRAC as well as MIPS data (Backus et al. 2005; Velusamy et al. 2008). The deconvolution has been carried out with 50 iterations on the fully reduced SINGS MIPS mosaic at 24 μm and on the PAH IRAC 8 μm maps, using the point response functions provided by the Spitzer Science Center. The resulting HiRes deconvolved images are shown in Figure 1, along with the initial SINGS images at 24 μm (top left and top right, respectively) and 8 μm (bottom left and bottom right, respectively). The HiRes deconvolution provides an FWHM in the range of 09–15 and 22–26 for the 8 μm and 24 μm images, respectively, as obtained by fitting Gaussian models to several point sources in the vicinity of M51. The spread in the FWHM is caused by a lack of very bright stars in the FOV. We adopt a final resolution of 11 × 13 at 8 μm and 23 × 24 at 24 μm.

The benefit of applying the HiRes deconvolution is improved visualization of spatial morphology by enhancing resolution and removing the contaminating sidelobes. Removing the diffraction lobes in the Spitzer images is more critical than resolution enhancement for our analysis of the radio/IR correlation. The sharper inner edges of the spiral arms become clearly visible in the HiRes image. The 24 μm HiRes map reveals in detail the filamentary structure in the interarm regions and allows us to separate some point sources which are blended in the lower resolution SINGS maps. More importantly, the sidelobes of the MIPS PSF have been removed around point sources. This is evident in the case of the companion and for strong point sources in the disk of M51a.

HiRes deconvolution is most effective in removing diffraction spikes in the case of isolated point sources located in a zero background. However, when the point sources are embedded in a high background (like the nucleus of M51a and bright point sources in the spiral arms), point sources show a residual ringing at a very low level of 0.2%–0.5%, although the diffraction spikes are removed. This level is significantly higher than the nominal diffraction residual around isolated bright point sources. For comparison the ring around the nucleus of M51b is weak at the 0.2% level. Therefore, the nucleus of M51a and the region around the nucleus of M51b will be masked in the correlation analysis applied in the following sections. We verified that the flux has been conserved during the deconvolution by comparing the total flux of the initial and deconvolved maps, as well as the flux within individual regions. For this second test, we used 13'' diameter apertures centered on bright H ii regions as defined by Kennicutt et al. (2007). The total flux and the fluxes within the apertures centered on the bright H ii regions are conserved, with a difference of about 0.7% between the maps and the rms noise level of the final HiRes maps are about 0.0067 MJy sr⁻¹ at 24 μm and 0.0069 MJy sr⁻¹ at 8 μm.

The Hα image used in our study is the Hα emission-line map with the underlying stellar continuum subtracted as presented by Calzetti et al. (2005). An average constant correction factor has also been applied across the image to correct for the contribution of the two [N ii] emission lines (Calzetti et al. 2005). It was obtained on 2001 March 28 at the 2.1 m KPNO telescope as part of the SINGS Legacy Survey. The measured FWHM of the PSF is 19 comparable to the resolution of our radio continuum data. In addition to standard reduction procedures, correction for the vignetting effect in the southern region of the FOV was applied to these data, by comparing flux measurement of stars in the vignetted side of the mosaic with the same stars in the clean side. Final flux calibration was carried out using standard star observations. The final map, without any further background subtraction applied, is shown in Figure 2. The median value of dust extinction in Hα is A_V ∼ 2.8 in $\hbox{H\,{\sc ii}}$ regions (Calzetti et al. 2005; Scoville et al. 2001) and A_V ∼ 2 in diffuse gas (Scoville et al. 2001) and it increases toward the center (Calzetti et al. 2005). Dust absorbs the UV and optical light of young stars and re-emit in IR. Therefore, dust extinction hampers greatly any SFR estimation from optical and UV wavelengths.

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Figure 2 presents our final SSC radio continuum maps. These high-resolution radio images of M51 show unprecedented details. Many structures are revealed from point sources in the spiral arms to faint extended filaments in the interarm and outer regions. While the general distribution of the 20 cm continuum, dominated by synchrotron emission, is fairly similar to the morphology seen in the 3.6 cm and 6 cm emission (a mixture of synchrotron and thermal radio emission), the contrast of individual structures is changing from one frequency to another.

The spiral arms are bright and well defined at all wavelengths, yet faint extended emission is hardly detected at 3.6 cm. Within the arms, compact bright sources are present at all three frequencies. They mostly correspond to H ii regions and SNRs. The interarm regions present filamentary structures as well as fainter extended smooth emission at 20 cm and 6 cm. Those structures are not seen at 3.6 cm, due to the higher noise of the 3.6 cm data and its intrinsically higher faintness as evidenced by its spectral index (α ∼ −0.9, with S_ν ∝ ν^α; Fletcher et al. 2010). Distinct 20 cm radio continuum emission is present between the northern arm of M51a and its companion. This region is less bright in the 6 cm map and is hardly evident in the 3.6 cm map. It is consistent with synchrotron emission that is enhanced by the ongoing interaction of NGC5195 with M51a (see E. Schinnerer et al. 2011, in preparation for a detailed analysis of this region). In particular, an interesting feature in this region is the arc-shaped structure south of M51b, also observed in Hα emission (Greenawalt et al. 1998; Figure 2). North of NGC5195, extended faint emission is present at 20 cm and 6 cm. In the 20 cm and 6 cm images, one background radio galaxy with strong radio lobes is observed east (R.A. =13^h30^m16^s; δ = +47°10'22'') of M51. This object is brighter at 20 cm than 6 cm, which implies a steep spectrum source. It is not seen in the 3.6 cm map since it lies outside the mosaic.

M51a hosts a type 2 active nucleus. The inner kiloparcsec region observed at 20 cm is shown in Figure 3. The nucleus is unresolved and located at R.A. =13^h29^m527; δ = +47°11'425 (J2000), in good agreement with the position reported by Crane & van der Hulst (1992) at 6 cm. In addition to the radio source, two main nuclear features are observed: a ring-like structure 5'' north of the nucleus, thought to be an outflowing bubble, related to the nuclear activity (Maddox et al. 2007) and bright compact emission south of the nucleus corresponding to the edge of the radio jet, identified by Crane & van der Hulst (1992) at 6 cm. The jet itself, 3'' long, is hardly resolved at our resolution (14 at 20 cm) as seen in Figure 3.

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Using the task IRING we calculated the total integrated flux in the primary beam corrected maps. We used rings centered on the active galactic nucleus (AGN) location and assumed a position angle of 170° and a disk inclination of 20°. We found total fluxes of 0.3 ± 0.2 Jy at 3.6 cm, 0.6 ± 0.2 Jy at 6 cm, and 1.4 ± 0.1 Jy at 20 cm (including the companion M51b, see Table 3). These fluxes are in good agreement with previous single dish and interferometric observation. Indeed at 3.6 cm Klein & Emerson (1981) derive a flux of 0.306 ± 0.026 Jy, at 6 cm Israel & van der Hulst (1983) report a flux of 0.525 ± 0.033 Jy, and at 20 cm Segalovitz (1977) give a flux of 1.5 ± 0.1 Jy. The global spectral index is α = −0.9 (where the radio flux is expressed as S_ν ∝ ν^α), in good agreement with the results of Fletcher et al. (2010).

Within each tilted ring, we also derived the azimuthal average radio flux. Figure 4 presents the radial profiles at 3.6 cm, 6 cm, and 20 cm, from 25 ≈ 100 pc to 400'' ≈ 16 kpc, encompassing the entire extent of M51. The azimuthal averaged radial profiles are similar at all three wavelengths. They show several peaks at the same location: 25'', 67'', 120'', and 147'' (1 kpc, 2.7 kpc, 4.8 kpc, and 5.9 kpc). Those radii correspond to the inner and outer spiral arms. One difference between our three maps is that the profile at 20 cm and 3.6 cm decreases for radii R >260'' ≈ 10.5 kpc and R >220'' ≈ 9 kpc, respectively, while at 6 cm the intensity remains constant at large radii. The radio profiles at 20 cm and 6 cm present a peak at about 250'' (10 kpc), which is not seen at 3.6 cm. This peak corresponds to NGC5195. It does not show up in the 3.6 cm radial profile due to increase of noise at these large radii in the 3.6 cm map.

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We fitted the radial profiles for radii between 50'' and 280'' (Figure 4) with exponentials of the form $I_0 \times \exp {(-\frac{r}{h})}$, where h is the exponential radial scale length of the profile. The corresponding scale lengths are 3.8 kpc ± 0.2 kpc at 3.6 cm, 5.6 kpc ± 0.4 kpc at 6 cm, and 4.5 kpc ± 0.5 kpc at 20 cm (see Table 3). Our values are lower than the scale length of the molecular gas disk (6.5 kpc; Hitschfeld et al. 2009) but the scale length at 6 cm is in good agreement with the dust scale length (5.4 kpc; Meijerink et al. 2005). At 3.6 cm the smaller scale length corresponds to the thermal emission, concentrated close to H ii regions. We find a smaller scale length at 20 cm than at 6 cm, suggesting that we may miss a very extended smooth component, consisting of old CR electrons, emitting synchrotron emission. Although our VLA data are theoretically sensitive to structures up to 15' (about 35 kpc), single dish observations seem to be needed to correctly map the largest scale disk at 20 cm. For comparison, NGC6946 presents a scale length of 4.0 kpc at 20 cm (NT emission) and 4.6 kpc at 6 cm (Walsh et al. 2002) while in M33, the scale lengths are much smaller: 2.7 kpc at 20 cm, 2.3 kpc at 6 cm, and 1.3 kpc at 3.6 cm (Tabatabaei et al. 2007a). The missing flux at 20 cm does not pose a problem for our wavelet analysis, as it only affects spatial scales larger than ∼400''. We verified this by comparing the wavelet analysis of the 6 cm images with and without short spacing correction.

To apply the wavelet analysis to our data, we first convolve our radio and IR maps to a common resolution of 24 (which is the resolution of our HiRes 24 μm and radio 3.6 cm images). The maps were regridded to the same geometry with a spatial sampling of 09 and centered at R.A. =13^h29^m527; decl. =+47°11'425 (J2000), the location of the nucleus of M51a. Finally, we cut the images to a common FOV of 12' × 12' (799 × 799 pixels). We subtracted five bright radio sources as they dominate the wavelet spectra at small spatial scales and therefore mask the contribution of fainter features. These sources are the two galactic nuclei (including the jet and bubble related to the AGN of M51a and the surrounding of the nucleus of M51b) and three luminous radio sources at the coordinates R.A. =13^h29^m305; decl. =+47°12'505, R.A. =13^h29^m516; decl. =+47°12'08'' and R.A. =13^h30^m05^s; decl. =+47°10'357 (J2000). The latter three sources correspond to the sources 1, 37, and 104 in the catalog of Maddox et al. (2007). As they exhibit steep radio spectral indices and no optical or X-ray counterparts have been detected, they are very likely background radio galaxies (Maddox et al. 2007).

In order to avoid any boundary conditions while scanning the dynamical range of spatial scales with the wavelet decomposition, the final maps are three times larger than the original maps (2397× 2397 pixels). We developed a new tool to calculate the wavelet transform using MatLab and the YAWTb toolbox (Yet Another Wavelet Toolbox⁶), to be able to handle such large maps. Within this toolbox, the continuous two-dimensional wavelet transform is implemented in terms of Fourier transforms, using the FFT algorithm within MatLab:

$$\begin{equation} W(a,\boldsymbol{x}) = \int _{-\infty }^{\infty }\int _{-\infty }^{\infty } \hat{f}(\boldsymbol{k}) \hat{\psi ^{*}}(a\boldsymbol{k})e^{2i\pi \boldsymbol{k}.\boldsymbol{x}} d\boldsymbol{k}, \end{equation} \tag{ 1 }$$

where f is the two-dimensional function to be analyzed and ψ is the wavelet function. $\hat{f}$ and $\hat{\psi }$ are the Fourier transforms of f and ψ, respectively.

The choice of the analyzing wavelet depends on the data and the goals of the analysis. Here we choose the Pet Hat function introduced by Frick et al. (2001) which allows a better separation of scales than most commonly used wavelets (e.g., Morlet, Haar or Mexican Hat wavelets). The Pet Hat function is defined in the Fourier space as

$$\begin{equation} \hat{\psi }(\boldsymbol{k}) = \left\lbrace \begin{array}{@{}ll}\cos ^2 \left(\frac{\pi }{2}\log _2 \frac{k}{2\pi }\right), & \pi \le k \le 4\pi \\ 0, & k> 4\pi, k<\pi \end{array} \right., \end{equation} \tag{ 2 }$$

where $k=|\boldsymbol{k}|$.

While the wavelet analysis is robust to noise, compared to Fourier analysis (Frick et al. 2001; Tabatabaei et al. 2007b), it is not obvious how exactly noise affects the smallest scales probed at low signal-to-noise ratio (S/N). In our radio data, the noise at the map edges is increased due to the primary beam correction of the interferometric observations. Since we have added single dish data to our 6 cm and 3.6 cm VLA maps for short spacing correction, blanking pixels of values lower than, for example, 2σ_rms removes also a lot of this intrinsically faint, but real, emission. This is not the case for the 20 cm and the IR and optical maps, but for consistency we applied the wavelet analysis to our data without blanking pixels with low S/N.

We tested the influence of noise by gradually adding Gaussian noise to our 20 cm map (which has the best S/N of the radio data) from 11 μ Jy beam⁻¹ (noise level of the 20 cm map) to 25 μ Jy beam⁻¹ (noise level of the 3.6 cm map) in steps of 1 μ Jy beam⁻¹. The description of the noise effect on the wavelet analysis can be found in Appendix B. The conclusion of our test is that noise affects the wavelet analysis on small spatial scales: a < 18'' (a < 10'') at the noise level of the 3.6 cm map (the 6 cm map, rms ≈ 16 μ Jy beam⁻¹). Therefore, we consider the wavelet spectra of 3.6 cm and its correlation with the other wavelengths only on scales larger than 18'', while this limit is set to 10'' for the spectrum and cross-correlation of the 6 cm data.

For the final analysis, we decompose the radio, IR, and Hαmaps into 50 spatial scales between 5'' (twice the resolution, except for the radio maps at 3.6 cm and 6 cm, see above) and 650'' (≈26 kpc, maximum extent of M51a) to encompass all structures within the galaxy with enough spatial sampling to compare the morphology at different wavelengths.

In this section we describe the results of the wavelet analysis of M51a's radio, IR, and optical maps. Table 4 summarizes the interesting scales found in the wavelet spectra (labeled with an "S," second and third columns) or cross-correlation between two images (labeled with an "X," fourth and fifth columns). By construction, the steps between the spatial scales are about 10% of a: $\delta a \sim \frac{a}{10}$, increasing toward larger scales. Moreover as it is described in details in the following, there are shifts of the interesting scales between the wavelengths. Therefore, for a better clarity of Table 4, the errors on a listed in the first column are the maximum between the steps and the spatial shift of the local extrema discussed below between the different emissions.

4.2.1. Wavelet Spectra

The wavelet spectrum, M(a), represents the distribution of the emitting power as a function of the spatial scale a:

$$\begin{eqnarray*} &&M(a) = \int _{-\infty }^{\infty }\int _{-\infty }^{\infty } |W(a,\boldsymbol{x})|^2 d\boldsymbol{x}, \end{eqnarray*}$$

where $W(a,\boldsymbol{x})$ is the wavelet decomposition map at the spatial scales a (see Frick et al. 2001, and Appendix A.2 in this paper for a detailed mathematical definition). Toward larger scales, a smooth increase in M(a) is expected if most of the emission is coming from diffuse structures, and a decrease if strong point-like or compact structures are present. Peaks in the distribution of M(a) indicate dominant scales in the morphology.

The wavelet spectra derived for various continuum and line maps of M51 are presented in Figure 5. For a better clarity of this figure, the spectra have been scaled by powers of 10, avoiding any confusion. None of the spectra increases smoothly toward larger scales indicating the absence of a dominant diffuse disk. The only general increase is visible at 20 cm, while the wavelet spectrum at 24 μm decreases and the spectra of the other frequencies remain roughly flat. These general trends are accompanied by some fluctuations, which indicates that prominent scales, and thus distinct morphological structures, exist. Local maxima and minima in the wavelet spectra of our different maps are listed in Table 4, in the second and third columns, respectively.

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All three wavelet spectra of the radio data exhibit similar substructures. The 3.6 cm (6 cm) spectrum starts at 18'' (10'') as the noise dominates at smaller scales (see Appendix B). The fluctuations in the three radio spectra can be interpreted as several local extrema, roughly located at the same spatial scales. All three radio wavelet spectra show a small bump at a ≃ 20'' (0.8 kpc, corresponding to complexes of star-forming regions and structures within the spiral arms). The second local maximum can be attributed to the width of the spiral arms seen in radio and a third maximum appears on a larger scale where the resolved structures include the inner disk of M51a, NGC5195, and the outer spiral arms. Interestingly, these two maxima appear at slightly larger scales at 20 cm (respectively at a ≃ 55'' ∼ 2.2 kpc and 150'' ∼ 6 kpc) than for the other two radio wavelengths (at about 45'' ∼ 1.8 kpc and ∼135'' ∼ 5.5 kpc), indicating wider spiral arms and a larger disk at 20 cm likely due to the propagation of CR electrons.

The most dominant scale in our data is found at a = 10'' (0.4 kpc) where the 24 μm emission is considerably strong. No dominant scale is seen in the 8 μm (PAH) or the Hα emission. The latter could indicate a strong effect of extinction in the Hα emission by dust flattening the spectrum on small scales (since we would expect a small-scale dominance in the Hα emission, similar to the one found for the 24 μm emission). Small scales (a< 20'') contribute more to the emission at the IR and optical wavelengths than in the radio continuum. Although the IR emission at 8 μm and 24 μm is mostly related to warm dust, their spectra are strikingly different. While the 24 μm emission is strongest at 10'', the PAH 8 μm spectrum is approximately flat over our spatial scales range. This reflects the fact that the 24 μm emission is more strongly peaked on star-forming regions than the PAH 8 μm. The 8 μm spectrum reaches a very weak local maximum at about 16'' ∼640 pc. This weak maximum could trace the PAHs illuminated by young and massive stars. However, PAHs emission arising from PDRs surrounding the H ii regions (Helou et al. 2004; Murphy et al. 2006; Bendo et al. 2006; Gordon et al. 2008) is not predominant and the contribution of PAHs heated by the interstellar radiation field on larger scales is significant, as shown in previous studies (e.g., Bendo et al. 2008). Moreover, dust emission may be important in regions of the intense radiation field at 8 μm (Draine & Li 2007) which could also explain the significant contribution of the large scales in the wavelet spectrum of this wavelength. The 8 μm spectrum shows a local maximum at a ≃ 45'' (∼1.8 kpc) similar to the radio 3.6 cm and 6 cm emission spectra (the wavelet spectrum at 20 cm reaches a maximum at 55''). The Hα and 24 μm wavelet spectra present no minimum at this scale and decrease smoothly out to a ≃ 200'' (∼8 kpc). This indicates that while structures corresponding to the spiral arms are clearly visible at 8 μm they do not dominate the morphology of the Hα and 24 μm emission. Both IR and optical wavelet spectra have a weak maximum around scales of 120'' –130'' (∼4.5 kpc) corresponding to structures encompassing M51a inner disk, the outer spiral arms and the companion NGC5195. Finally, a strong minimum at 200''–220'' (∼8.4 kpc) is present in both IR and optical wavelet spectra at the same location than the minima in the radio wavelet spectra. This is moreover the only obvious feature in the Hα spectrum. This strong local minimum in all wavelet spectra may reflect the good contrast between the disks of the two interacting galaxies on this spatial scales (∼8.4 kpc). This minimum is also weaker in the wavelet spectra of the radio emission, revealing a diffuse radio structure encompassing both M51a and M51b and interestingly the 3.6 cm wavelet spectrum reaches this local minimum at larger scales (∼300'' ∼ 12 kpc). The locations of the local extrema in the optical and IR wavelet spectra are similar to those of the extrema present in the radio spectra, albeit at systematically shorter spatial scales: the sizes of the corresponding structures are 10% smaller than those in the radio maps. This shift between the sizes of optical/IR and radio structures is likely reflecting the diffusion of the CR electrons into the ISM which makes the radio structures appear wider and larger than the IR and optical counterparts.

Finally, at all wavelengths the wavelet spectra reach a maximum around 600'' (∼24 kpc) and remain almost constant at this maximum for larger scales. On these scales, the wavelet probes the extended galactic disk of M51a. This maximum is reached at smaller scales in the IR and Hα than in the radio maps, the difference is about 5 kpc. This shows a larger extent of the disk in radio than in IR and Hα, again indicating propagation of CR electrons producing the radio emission. This finding is in good agreement with Murphy et al. (2006, 2008) who showed that radio images can be interpreted as a smeared version of the IR maps.

4.2.2. Wavelet Cross-correlations

The wavelet spectrum allows us to identify the dominant energy structures in our data and to qualitatively compare them. The wavelet cross-correlation is a useful method to compare different images as a function of spatial scales as shown in several studies (e.g., Frick et al. 2001; Hughes et al. 2006; Tabatabaei et al. 2007b). Indeed, while standard cross-correlation analysis, such as a pixel-to-pixel correlation, can be dominated by bright extended regions or large-scale structures, the wavelet cross-correlation r_w allows for the analysis of a scale-dependent correlation between two images. It is also very sensitive to any differences between two images even on spatial scales where the wavelet spectra appear similar. Such differences could be, e.g., an azimuthal offset between spiral arms or a systematic shift of structures of similar size. The definitions and equations used here are detailed in Appendix A.3. Following Frick et al. (2001) we consider the correlation between two images to be significant for values higher than r_w>0.75. Fourth and fifth columns of Table 4 list the local maxima and minima in the wavelet cross-correlation of our different maps.

There is significant correlation between the radio images at all scales larger than ∼20'' (Figure 6). At all scales, the best correlation occurs between the 20 cm and 6 cm maps while the weakest is between 20 cm and 3.6 cm particularly for scales a < 140''. Since the observed radio continuum is a combination of thermal (free–free) and NT (synchrotron) emission, the contributions from thermal and NT emission change as a function of wavelength as well as location in the galactic disk, the fraction of NT emission being more and more important at long wavelength as well as on large structures. This can explain the differences between the wavelet correlations shown in Figure 6. Two local maxima and two local minima are present in these cross-correlations at similar spatial scales and are especially pronounced in the cross-correlations involving the 3.6 cm data. The locations of the first local maximum and minimum correspond well to those seen in the wavelet spectra of the radio maps and corresponds to the width of the spiral arms (about 2 kpc) and the size of the structures formed by the combination of the spiral arms and the interarm region (∼3.5 kpc). The local minima could be attributed to different slopes and displacements in the extrema of the corresponding wavelet spectra (Figure 5 and Table 4) again caused by more diffuse morphology of the longer wavelength radio emission (due to a higher contribution from synchrotron emission).

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Now we study the wavelet correlation between the radio continuum at 20 cm, 6 cm, and 3.6 cm and the IR/optical emission (Figures 7 and 8, respectively). First, we note that all the correlation coefficients decrease globally toward smaller scales and this decrease is particularly steep from a = 40'' downward, which corresponds to the width of the spiral arms. The correlation of the IR emission at 24 μm is higher with the radio continuum at 3.6 cm than at 6 cm and 20 cm on spatial scales smaller than 100'' (Figure 7). This is expected, as the radio continuum at 3.6 cm contains less synchrotron contribution than at 6 cm or 20 cm, the high correlation between 3.6 cm and IR data reflects the common origin of the thermal radio and IR emission, both being powered by ionizing young stars in the H ii regions. Local minima and maxima are present in the three correlations between radio and IR emission, appearing at similar spatial scales as the local extrema in the cross-correlation between the radio wavelengths themselves. Finally, the correlation between 24 μm and 8 μm PAH emission is the strongest correlation involving IR or optical emissions, at all spatial scales. It reaches a weak local minimum at a = 245'' (∼7.3 kpc) which is the location of the local maximum of the cross-correlation between 24 μm and 3.6 cm.

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The correlations including Hα emission decrease more rapidly than the corresponding correlations with IR emission (Figure 8). They all present a local maximum at about 135'' (∼5.7 kpc) and minimum at 200'' (∼8.2 kpc, this local minimum is reached at slightly larger scale for the correlation with 3.6 cm). The wavelet cross-correlations between Hα and the IR emissions at 24 μm and 8 μm are very similar. As in the case of the 24 μm emission (see Figure 7), the wavelet cross-correlations between Hα and the radio emissions are lower at 20 cm than at 6 cm and 3.6 cm at all scales a < 200''.

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Here, we discuss important spatial scales in M51a galactic disk, as revealed by the wavelet spectra and cross-correlations, and their physical implications. Figure 9 shows the original maps smoothed to 24 resolution at 20 cm, 6 cm, 3.6 cm, 24 μm, and 8 μm and Hα from top to bottom, and the corresponding wavelet decomposition maps at four prominent spatial scales: 10'' (450 pc), 45'' (1.8 kpc), 83'' (3.3 kpc), and 200'' (8 kpc). The wavelet decomposed maps show the structures which contribute at the different scales. The wavelet maps at 10'' (Figure 9, second column) show small structures which are dominating the wavelet spectrum at 24 μm (see Figure 5). Those features also contribute largely to the 8 μm and Hα emission, although their spectra do not show a pronounced peak at this scale. They are formed of compact structures (especially in Hα) corresponding to H ii regions and filament-like features (especially at 8 μm).

At 45'' (∼1.8 kpc) we distinguish elongated structures defining the spiral arms (Figure 9, third column). This spatial scale is of the order of the typical width of the spiral arms of M51a. At this spatial scale the cross-correlations between the radio/IR/optical data reach a local maximum (Section 4.2.2). It is interesting to note that some filamentary structures in the interarm regions contribute to the energy at this spatial scale, the length of these filaments being of the same order of the width of the spiral arms.

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Finally, at 200'' (8 kpc), the wavelet decompositions show the diffuse emission from the galactic disk and the region between M51 and its companion (northeast). This interaction zone dominates at long radio wavelengths (20 cm and 6 cm), it is weaker at 3.6 cm and disappears in the IR and optical maps. This spectral trend indicates that the emission in this region is mainly due to NT radiation, such as synchrotron emission. The spatial scale corresponds also to a local minimum in the radio correlations between Hα emission and the radio and IR emissions, likely due to this extra radio component in the interaction region (Figure 8).

As shown in the previous sections, wavelet analysis is a powerful tool to determine the dominant morphological structures in our multi-wavelengths images. In the case of M51a, the scale at a = 83'' ≈ 3.5 kpc is particularly interesting. It corresponds to a local minimum in the wavelet spectrum of all radio and IR images (Figure 5), which appears deeper at low than high radio frequency, as well as in the correlation between the radio and IR maps (see Section 4.2.2). This characteristic is also seen in NGC6946 (Frick et al. 2001) and M33 (Tabatabaei et al. 2007b) at roughly the same spatial scale, 3.5 kpc. We interpret this minimum due to the best contrast between the arm/interarm region. Indeed at this scale, the positive part of the wavelet function filters the structures formed by the combination of the bright spiral arms and the "darker" interarm regions. The resulting wavelet coefficients contain then information from these two regions and on the spatial scale where the contrast between spiral arms and interarm is optimal, the wavelet coefficients are then lower than at smaller or larger scales, for which the contrast is less good and then more energy is contained into the positive part of the wavelet function. This leads to a local minimum in the wavelet spectrum at the corresponding spatial scale. Thus, the wavelet transforms at the scale of 83'' of M51a's radio and IR images trace well the spiral arm structure (see Figure 9, fourth column). We will use this result to investigate the local radio–IR correlation in M51a in the following two sections. First, we discuss the wavelet cross-correlation between the 20 cm and 24 μm maps, then we use the wavelet transform at 83'' to define different regions in M51a's galactic disk (spiral arms, interarm region, outer and inner regions) and discuss the correlation between 20 cm and 24 μm as a function of the environment.

The local radio–IR correlation is formed on one hand by the strong relation between the thermal radio (Bremstrahlung) and the warm dust emission present in H ii regions, and by the, albeit weaker, correlation between the synchrotron emission and the cool dust (Hoernes et al. 1998). Here, we discuss the correlation between the 20 cm radio continuum, which is the most sensitive tracer for synchrotron radiation in our radio data (Condon 1992) and the 24 μm emission within M51a. We use the 24 μm emission to investigate the radio–IR correlation since it has the same spatial resolution as our radio maps, although it is mostly related to warm dust. Synchrotron and IR continuum exhibit many complex structures over all spatial scales as shown in Figures 1 and 2. Moreover they do not always arise from the same location, and in some regions, NT radio emission is not at all associated with IR emission, as in the interaction region between M51a and NGC5195. CR electrons are usually produced and accelerated in SNRs and radiate synchrotron emission while propagating in the magnetized ISM. In the regions where radio emission is not spatially associated with IR, synchrotron emission from CR electrons may be enhanced (e.g., by strong magnetic field) or produced by a different physical process (such as galactic-scale shocks) rather than SF-induced shocks.

The wavelet spectra reveal that the predominant structures at 20 cm and 24 μm are different (see Section 4.2.1). While small-scale structures dominate the IR emission, they carry only little energy in the radio continuum maps (Figure 5). Those structures correspond to bright knots in the spiral arms, associated with H ii regions and stellar clusters (Figure 9). The spiral arms at 20 cm are wider than at the other wavelengths (∼60'' compared to about 40'' in 3.6 cm and IR maps as traced by the maximum in the wavelet spectra, see Section 4 and Table 4), reflecting the diffuse synchrotron emission extending into the interarm regions. On larger scales (83'' ≈ 3.5 kpc, see Figure 7) the correlation between 20 cm and 24 μm emission reaches a local minimum. The two wavelet transform maps at these spatial scales show similar structures, tracing the spiral arms and interarm region (Figure 9, fourth column). However, an obvious difference between these two maps is that the spiral arm structure revealed by the wavelet transform of the 20 cm emission at this spatial scale is more continuous than the structures tracing the arms in the 24 μm wavelet transform. This could be explained by the diffusion of the CR electrons from their acceleration site into the ISM.

The spatial scale on which the correlation between the NT–radio continuum and the IR emission should break is generally assumed to be the diffusion length of CR electrons. Following our wavelet analysis, the correlation between 20 cm and 24 μm starts to decrease rapidly from scales of 45'' (1.8 kpc) downward. Between 30'' and 20'' (1.2–0.8 kpc) the correlation coefficients get stabilized around a value of 0.54 and then decrease further from 0.5–0.8 kpc toward smaller spatial scales, in good agreement with CR electrons' diffusion lengths of about 0.5 kpc in M51 estimated by Murphy et al. (2006, 2008). This shows that the radio–IR correlation is not smooth and exhibiting local maxima and minima at different spatial scales corresponding to particular morphological structures in the galaxy. This result reflects the complexity of the local radio–IR correlation and hints that different processes lie at the origin of this relation and that while one process may dominate at a particular spatial scales, none is dominating at all scales. In particular, the fraction of NT and thermal emission can change with spatial scales, while diffuse IR emission form older stellar population can contaminate the emission at 24 μm considered here. These different components of the radio and IR relation can explain the observed variations in the radio/IR relation (traced here by the correlation between the 20 cm and 24 μm emission). This will be developed in details in a further study, where in particular careful decomposition of the radio continuum into thermal and NT components will be done (G. Dumas et al. 2011, in preparation). The decrease of the correlation coefficients toward smaller spatial scales can be interpreted as a break of correlation between the radio and IR emission on those scales. In particular, it is interesting to note that the correlation between the 20 cm and 24 μm emission does not hold on typical scales of star-forming complexes. The break of correlation on these small scales is expected given the timescales difference between the IR and NT radio emission and the diffusion length of CRs in the ISM.

In order to investigate the radio–IR correlation in the different environments of M51a, we identified different regions within the galactic disk. The spiral arms and interarm region are defined by the wavelet transform at the spatial scale a = 83'', as this spatial scale provides the best contrast between the arm/interarm region (see Section 4.2.1). Positive values of the wavelet filtered 20 cm image at 83'' define the spiral arms, while negative values mask the interarm region. The region with R < 1.75 kpc is the inner region and the outer disk region starts at the outer edges of the external spiral arms, at roughly R >6.5 kpc. We excluded the companion NGC5195 from the analysis. The different regions are filled with apertures of 6'' diameter. Figure 10 shows the different regions overlaid on the 20 cm map. The 20 cm and 24 μm flux densities within these apertures are calculated with the task "aphot" in IRAF. Figure 11 presents the flux densities at 24 μm and 20 cm in the spiral arms (top left panel), interarm (top right panel), outer disk (bottom left panel), and inner region (bottom right panel).

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We fitted the relation between the 24 μm and 20 cm emission with the function $F_{24\mu \rm m} = b \times (S_{20\rm cm})^c$, with $F_{24\mu \rm m}$ and $S_{20\rm cm}$ being the flux density at 24μm and at 20 cm (in Jy), respectively. The fitted values and standard deviations of the scaling exponent c and the coefficient b are listed in Table 5. Prior to this fit, some points in the different regions were excluded. Those points are marked in Figure 12. They correspond to AGN-related emission (in particular the radio jet) and bright H ii regions in the inner region (white circles in Figure 12), to bright H ii regions in the spiral arms and interarm region (respectively, red and blue circles in Figure 12) and to one point at very low 24 μm flux density in the outer region (green circle in Figure 12). Those outliers are also highlighted as green squares in Figure 11. The results of the fit in each region are given in Table 5 and plotted as red lines in Figure 11.

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Table 5. Results of the Fit: $F_{24\mu \rm m} = b (S_{20 \rm cm})^c$

Fit Parameters	Spiral Arms	Interarm	Outer	Inner	Total
〈c〉	1.01	0.51	0.48	1.5	1.00
σc	0.01	0.01	0.01	0.1	0.01
〈log b〉	0.9	−0.50	−1.19	2.3	0.81
σlog b	0.01	0.01	0.02	0.3	0.03
〈q24〉	1.01	0.91	0.63	1.0	0.86
$\sigma _{q_{24}}$	0.2	0.16	0.17	0.1	0.3

Notes. Means and dispersions of the fit parameters b and c for the different regions, assuming $F_{24\mu \rm m} = b (S_{20\rm cm})^c$ (fluxes derived within 6'' apertures). The means and dispersions of the q₂₄ parameter from Murphy et al. (2006) are also listed for comparison.

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The relation between $F_{24\mu \rm m}$ and $S_{20\rm cm}$ is linear in the spiral arms and globally over the galaxy. In this case, the logarithm of parameter b is equal to the parameter q₂₄ defined by Murphy et al. (2006) as $q_{24}\equiv \log [ {{F_{24\mu \rm m} (\rm Jy) \over S_{20\rm cm} (\rm Jy)}}]$ and we find that our fitted parameter is in good agreement with the values derived by Murphy et al. (2006) (see Table 5). In the interarm region and outer disk, the flux density at 24 μm is proportional to about the square-root of the 20 cm flux density while in the inner regions we observe an opposite behavior: $F_{24\mu \rm m} \propto (S_{20\rm cm})^{1.5}$. In these regions where a nonlinear relation occurs between $F_{24\mu \rm m}$ and $S_{20\rm cm}$, q₂₄ is determined by both log(b) and c, hence is not constant and of a limited use as a quantitative measure of the radio–IR correlation.

The large differences in the exponent 〈c〉 of the infrared–radio correlation (Table 5) demonstrate that the physical processes involved are different in the arm, interarm, and inner regions, while the interarm and outer regions show a similar behavior. As the spiral arms dominate the total flux emission (50% at 20 cm and 60% at 24 μm), a linear behavior is obtained when looking at the whole galaxy, as observed for the integrated emission of most galaxies (Yun et al. 2001; Bell 2003). Our results are especially striking in view of the many processes involved: star formation, dust heating, dust opacity, CR acceleration, CR propagation, and magnetic fields (see, e.g., Bicay & Helou 1990; Murphy et al. 2006). In spite of the complicated interplay of the ISM's different components, three well-defined regimes can be distinguished and are discussed in detail below.

Here, we are following the formalism and assumptions used by Niklas & Beck (1997). Assuming equipartition between the energy densities of the magnetic field and CR electrons, we can couple the radio NT flux density to the gas volume density. The energy density equipartition assumes that the energy densities of CRs (electrons and protons) and magnetic field are approximately equal and gives

$$\begin{eqnarray*} &&S_{20\rm cm}\propto B_{\mbox{eq}}^{3+\alpha }, \end{eqnarray*}$$

where B_eq is the total equipartition magnetic field strength (Beck & Krause 2005) and α is the synchrotron spectral index. The equipartition magnetic field strength B_eq is also referred to as the minimum energy magnetic field as it is the minimum value of B to realize this energy equipartition. This assumption has been and is still discussed (Niklas & Beck 1997; Beck & Krause 2005; Thompson et al. 2006). In particular, Thompson et al. (2006) compare this value of the magnetic filed strength measured in different galaxies to different scaling relations between B and the gas surface density and show that the equipartition magnetic field strength underestimates the magnetic field in starburst galaxies. M51 however follows the relation of normal star-forming galaxies for which the magnetic energy density of the equipartition magnetic field is comparable to the total hydrostatic pressure of the ISM (see Equation (3) and Figure 1 of Thompson et al. 2006). In that case we can assume that the equipartition magnetic field strength reflect to total magnetic field in M51 and that energy density equipartition is a good assumption for this galaxy. In M51, Fletcher et al. (2010) find α ≃ 0.6 in the spiral arms and inner region, and α ≃ 1.2 in the interarm and outer regions. In addition, the magnetic field is coupled to the gas:

$$\begin{eqnarray*} &&B\propto \rho _g^{\beta }, \end{eqnarray*}$$

where β describes the coupling between the magnetic field and the gas volume density ρ_g. For turbulent field amplification (e.g., by dynamo action), β ≃ 0.5 is expected (Beck et al. 1996; Thompson et al. 2006). This value also emerges from this time equipartition between magnetic and turbulent energy densities. A similar value is predicted for cloud collapse along magnetic field lines (Mouschovias 1976), while isotropic collapse in turbulent fields gives β ≃ 2/3. In case of one-dimensional compression or shear of magnetic fields by large-scale flows of diffuse gas, β ≃ 1 is expected.

We therefore obtain a relation between the radio NT emission and the gas volume density:

$$\begin{eqnarray*} &&S_{20\rm cm}\propto \rho _g^{\beta (3+\alpha)}. \end{eqnarray*}$$

On the other hand, we can also link the 24 μm flux density to the gas density. In the case of optically thick dust for UV photons, the dust re-processes the totality of the radiation of the ionizing stars and thus the IR emission traces the star formation rate (SFR). Therefore we assume: $F_{24\mu \rm m}\propto I_0$ where I₀ is the energy density of the dust-heating radiation field and $I_0\propto \rm SFR$ where $\rm SFR$ is the star formation rate, hence $F_{24\mu \rm m} \propto \rm SFR$. Moreover, the Schmidt law (Schmidt 1959) gives a relation between the SFR and the gas volume density ρ_g: $\rm SFR \propto \rho _g^{n}$, with n being the Schmidt law index. Therefore we can write

$$\begin{eqnarray*} &&F_{24\mu \rm m}\propto \rho _g^{n} \end{eqnarray*}$$

and n = 1.4 ± 0.3 (Niklas & Beck 1997). In that case, the correlation between $F_{24\mu \rm m}$ and $S_{20\rm cm}$ with the exponent c yields

$$\begin{equation} \beta ={\displaystyle {n \over c(3+\alpha)}}. \end{equation} \tag{ 3 }$$

If the dust is not optically thick for UV photons, as, e.g., in the galaxy M31 (Hoernes et al. 1998), then the optical depth of the dust τ has been taken into account:

$$\begin{eqnarray*} &&F_{24\mu \rm m} \propto \tau I_0. \end{eqnarray*}$$

I₀ is the energy density of the dust-heating radiation field directly proportional to the SFR, and τ ∝ ρ_dust, then assuming a constant dust-to-gas ratio we have a relation between the 24 μm IR flux and the gas:

$$\begin{eqnarray*} &&F_{24\mu \rm m} \propto \rho _g^{(1+n)} \end{eqnarray*}$$

and Equation (3) becomes

$$\begin{equation} \beta = {\displaystyle {n+1 \over c(3+\alpha)}} \end{equation} \tag{ 4 }$$

As shown in Calzetti et al. (2005), in most regions of M51, (e.g., the spiral arms) dust attenuation is high enough that the IR emission traces well the SFR and Equation (3) can be applied. However, in region where dust extinction is lower and diffuse IR emission, coming from dust heated by the old stellar component (the so-called Cirrus emission) dominates, as at large radii or in the interarm region, IR emission is not totally related to recent star formation. The IR emission for a fixed SFR will be lower in such regions. And in the case of diffuse Cirrus emission we have the following relations: F_cirrus ∝ ρ_dust ∝ ρ_g, where ρ_dust and ρ_g are the volume density of dust and gas, respectively (we assume a constant dust-to-gas ratio). Then, $F_{24\mu \rm m}\propto \rho _g$ and Equation (3) becomes

$$\begin{equation} \beta = {\displaystyle {1 \over a(3+\alpha)}}. \end{equation} \tag{ 5 }$$

We have derived three possible relations between the IR and 20 cm emission, depending on the properties of the ISM. We now discuss the three trends found between the radio and IR emission in the spiral arms, interarm region, outer disk, and inner region of M51a:

• 1.  
  In the spiral arms, we can assume the dust to be optically thick and directly apply Equation (3). Therefore, with 〈c〉 = 1.01 ± 0.01, α = 0.6 ± 0.02 (Fletcher et al. 2010), and n = 1.4 ± 0.3, we obtain β = 0.4 ± 0.1 which is slightly below the range of theoretical predictions, but with an overall good agreement. The expected value under the assumption that magnetic density energy is comparable to the turbulent energy density of the ISM is β = 0.5 (Beck et al. 1996; Thompson et al. 2006). Figure 1 of Thompson et al. (2006) shows that the equipartition magnetic field strength in M51 realizes this assumption., and in that case, the Schmidt law index should be n = 1.8 ± 0.2.
• 2.  
  In the interarm and outer region, the spectral index is higher α = 1.2 ± 0.06 (Fletcher et al. 2010). Therefore, with 〈c〉 = 0.51 ± 0.01, and n = 1.4 ± 0.3, we obtain β = 0.7 ± 0.1 in the case of optically thick dust. Assuming UV-thin dust for the interarm and outer regions, we obtain β = 1.0 ± 0.1, as expected for compression or shear of magnetic fields in flows of diffuse gas. This possibility should be further investigated by comparing high-resolution data in radio continuum, H i and CO.In addition, the Cirrus IR emission may provide another interpretation of the nonlinearity between $F_{24\mu \rm m}$ and $S_{20\rm cm}$ in the interarm and outer regions. In this region, we find that the radio–IR correlation is sublinear, which implies that for a fixed SFR the 24 μm is lower that predicted (assuming 20 cm traces the SFR), as expected for regions dominated by Cirrus IR emission. Applying Equation (5) we then find β ≃ 0.5 in the interarm and outer regions, as in the spiral arms, which is in good agreement with theoretical values from turbulent field amplification (Beck et al. 1996). Therefore, diffuse IR emission emitted by dust heated in large part by older stars seem a good explanation for the sublinear radio–IR correlation found in the interarm and outer regions.Further high-resolution observations and measurements of dust opacity and separation of the cold and warm dust components are needed to distinguish between these two possibilities.
• 3.  
  For the inner region we found 〈c〉 = 1.5 ± 0.1. In this region, the AGN related radiation, in particular the radio jet, has been excluded for the fit. With α ≃ 0.6 (Fletcher et al. 2010) and n = 1.4 ± 0.3 we obtain β = 0.26 ± 0.05, in the case of optically thick dust. This value of β is well below the range of theoretical predictions. The equipartition field is too weak compared to the gas density. Either the process amplifying the magnetic field is less efficient in the inner region, or the equipartition assumption yields field strengths which are too low. The former case needs further investigation which is beyond the scope of this paper. The latter case is known to be valid for strong synchrotron or inverse Compton losses of the CR electrons (Beck & Krause 2005). As the magnetic field and the radiation field have large energy densities in the inner region of M51a, the radio synchrotron emission is probably suppressed, hence the radio-derived SFR underestimates the true amount of star formation present in the inner 1.5 kpc.

In summary, we can explain the different exponents of the infrared–radio correlation seen in the different galactic environments by a change in the optical dust opacity and the relations between the magnetic field strength and the gas density and the presence of a "Cirrus" contribution to the IR emission. The steep exponent detected in the inner region is probably the effect of suppression of radio synchrotron emission due to energy losses of CR electrons. Future high-resolution observations of galactic disks with HERSCHEL and the EVLA (and later with the SKA) will provide an excellent tool to study the interactions of the ISM components in galaxies.

The wavelet analysis of an image is a generalization of the Fourier analysis, where the sinusoidal waves are replaced by a family of self-similar functions that depend on scale and location. These functions are generated by dilating and translating and rotating a mother function which is called the analyzing wavelet. Frick et al. (2001) showed that the wavelet analysis is well adapted for the investigation of scale-dependent properties in astrophysical objects. For this study, we restrict our analysis to the use of isotropic wavelets, we assume here that the structures that we will probe with such analysis are axisymmetric, hence with no favored orientation. We then do not apply any rotation to the mother wavelet. The continuous two-dimensional wavelet transform can be written as

$$\begin{equation} W(a,\boldsymbol{x}) = \omega (a) \int _{-\infty }^{\infty }\int _{-\infty }^{\infty } f(\boldsymbol{x^{\prime }}) \psi ^{*}\left(\frac{\boldsymbol{x^{\prime }}-\boldsymbol{x}}{a}\right) d\boldsymbol{x^{\prime }}, \end{equation} \tag{ A1 }$$

where a is the scale size, f is a two-dimensional function (the image), ψ* is the complex conjugate of the analyzing wavelet, $\boldsymbol{x}$ and $\boldsymbol{x^{\prime }}$ are coordinate vectors, and ω(a) is a normalization factor. In the following, this factor will be equal to $\frac{1}{a^2}$ for energy normalization between the different scales, the studied images, and the wavelet transforms at different scales being then in the same unit.

Following Frick et al. (2001) we defined the energy of the wavelet decomposition map at scale a as

$$\begin{equation} M(a) = \int _{-\infty }^{\infty }\int _{-\infty }^{\infty } |W(a,\boldsymbol{x})|^2 d\boldsymbol{x}, \end{equation} \tag{ A2 }$$

where $W(a,\boldsymbol{x})$ is the wavelet decomposition map at the spatial scales a.

The distribution of this energy with the scale a is the wavelet spectrum. This spectrum reveals the dominant scales in the initial image.

The cross-correlation is a useful method to compare different images. Frick et al. (2001) showed that the pixel-to-pixel cross-correlation is dominated by the large-scale structures. The information given by the wavelet transformations is used to probe the correlation between different wavelengths, for example, on the scales of the spiral arms, the interarm, and central regions. The wavelet cross-correlation coefficient at scale a is defined following Frick et al. (2001):

$$\begin{equation} r_w(a) = \frac{\int\!\! \int W_1(a,{\bm x}) W_2(a,{\bm x}) d{\bm x}}{\left[ M_1(a)M_2(a)\right]^{1/2}}, \end{equation} \tag{ A3 }$$

where W₁ and W₂ refer to the wavelet decomposition maps at scale a of two images of same size and geometry and M₁ and M₂ correspond to the wavelet coefficients at this scale (see Equation (A2)), and the uncertainty on this cross-correlation is defined as

$$\begin{equation} \Delta r_w(a) = \frac{\sqrt{1-r_w^2(a)} }{\sqrt{N-2}} \end{equation} \tag{ A4 }$$

with N = (L/a)², L being the linear size of the image. In this analysis we use r₂ = 0.75 as the arbitrary threshold for a significant correlation, following Frick et al. (2001) and Tabatabaei et al. (2007b).

The wavelet transform is conservative: there is no loss of energy when applying the wavelet transform to an image $f(\boldsymbol{x})$. We indeed have a Plancherel relation for the wavelet transform, similar to the one for the Fourier transform, in the case of an axisymmetric wavelet:

$$\begin{equation} ||f||^2=\int f({\bm x})^2d{\bm x} = {\displaystyle {2\pi \over C_\psi }}\int M(a) {\displaystyle {da \over a}}, \end{equation} \tag{ A5 }$$

where M(a) is the energy of the wavelet transform of f(x) at the spatial scales a and $C_\psi =\int { {\displaystyle {|\hat{\psi }({\bm k})|^2 \over k^2}} d{\bm k}}$ a constant, with $\hat{\psi }$ being the wavelet Fourier transform. This conservation equation is exact for continuous scales from 0 to infinity. Of course, in practice we need to choose a finite numbers of scales on which the wavelet transform is produce. However, Equation (A5) can be used to quantify the contribution of the structures between two spatial scales to the total energy.

As a consequence of the energy conservation, we have an exact reconstruction of the image f from its wavelet decomposition:

$$\begin{equation} f({\bm x})= {\displaystyle {2\pi \over C_\psi }}\int W(a,{\bm b}) \psi \left({\displaystyle {{\bm x}-{\bm b} \over a}}\right) d{\bm b}{\displaystyle {da \over a}} \end{equation} \tag{ A6 }$$

with W(a, b) being the wavelet transform of f(x), with respect to the wavelet ψ. As for the energy conservation equation, this reconstruction is mathematically exact but not realizable in practice since it requires an infinite number of scales from 0 to +∞. Image reconstruction (and compression) with wavelets uses the discrete wavelet transform, for which an exact image reconstruction can be achieved with a finite and discrete number of scales.